Example 54.17 Partial Proportional Odds Model

Cameron and Trivedi (1998, p. 68) studied the number of doctor visits from the Australian Health Survey 1977–78. The data set contains a dependent
variable, dvisits, which contains the number of doctor visits in the past two weeks (0, 1, or 2, where 2 represents two or more visits) and
the following explanatory variables: sex, which indicates whether the patient is female; age, which contains the patient’s age in years divided by 100; income, which contains the patient’s annual income (in units of $10,000); levyplus, which indicates whether the patient has private health insurance; freepoor, which indicates that the patient has free government health insurance due to low income; freerepa, which indicates that the patient has free government health insurance for other reasons; illness, which contains the number of illnesses in the past two weeks; actdays, which contains the number of days the illness caused reduced activity; hscore, which is a questionnaire score; chcond1, which indicates a chronic condition that does not limit activity; and chcond2, which indicates a chronic condition that limits activity.

Because the response variable dvisits has three levels, the proportional odds model constructs two response functions. There is an intercept parameter for each
of the two response functions, , and common slope parameters across the functions. The model can be written as

The test of the proportional odds assumption in Output 54.17.1 rejects the null hypothesis that all the slopes are equal across the two response functions. This test is very anticonservative;
that is, it tends to reject the null hypothesis even when the proportional odds assumption is reasonable.

The proportional odds assumption for ordinal response models can be relaxed by specifying the UNEQUALSLOPES option in the MODEL statement. A fully nonproportional odds model has different slope parameters for every logit i:

The nonproportional odds model is fit with the following statements. The TEST statements test the proportional odds assumption for each of the covariates in the model.

Selected results from fitting the nonproportional odds model to the data are displayed in Output 54.17.2.

Output 54.17.2: Results for Nonproportional Odds Model

Testing Global Null Hypothesis: BETA=0

Test

Chi-Square

DF

Pr > ChiSq

Likelihood Ratio

761.4797

24

<.0001

Score

957.6793

24

<.0001

Wald

688.2306

24

<.0001

Analysis of Maximum Likelihood Estimates

Parameter

dvisits

DF

Estimate

StandardError

WaldChi-Square

Pr > ChiSq

Intercept

0

1

2.3238

0.2754

71.2018

<.0001

Intercept

1

1

4.2862

0.4890

76.8368

<.0001

sex

0

1

-0.2637

0.0818

10.3909

0.0013

sex

1

1

-0.1232

0.1451

0.7210

0.3958

age

0

1

1.7489

1.5115

1.3389

0.2472

age

1

1

-2.0974

2.6003

0.6506

0.4199

agesq

0

1

-2.4718

1.6636

2.2076

0.1373

agesq

1

1

2.6883

2.8398

0.8961

0.3438

income

0

1

-0.00857

0.1266

0.0046

0.9460

income

1

1

0.6464

0.2375

7.4075

0.0065

levyplus

0

1

-0.2658

0.0997

7.0999

0.0077

levyplus

1

1

-0.2869

0.1820

2.4848

0.1150

freepoor

0

1

0.6773

0.2601

6.7811

0.0092

freepoor

1

1

0.9020

0.4911

3.3730

0.0663

freerepa

0

1

-0.4044

0.1382

8.5637

0.0034

freerepa

1

1

-0.0958

0.2361

0.1648

0.6848

illness

0

1

-0.2645

0.0287

84.6792

<.0001

illness

1

1

-0.3083

0.0499

38.1652

<.0001

actdays

0

1

-0.1521

0.0116

172.2764

<.0001

actdays

1

1

-0.1863

0.0134

193.7700

<.0001

hscore

0

1

-0.0620

0.0172

12.9996

0.0003

hscore

1

1

-0.0568

0.0252

5.0940

0.0240

chcond1

0

1

-0.1140

0.0909

1.5721

0.2099

chcond1

1

1

-0.2478

0.1743

2.0201

0.1552

chcond2

0

1

-0.2660

0.1255

4.4918

0.0341

chcond2

1

1

-0.3146

0.2116

2.2106

0.1371

Linear Hypotheses Testing Results

Label

WaldChi-Square

DF

Pr > ChiSq

sex

1.0981

1

0.2947

age

2.5658

1

0.1092

agesq

3.8309

1

0.0503

income

8.8006

1

0.0030

levyplus

0.0162

1

0.8989

freepoor

0.2569

1

0.6122

freerepa

2.0099

1

0.1563

illness

0.8630

1

0.3529

actdays

6.9407

1

0.0084

hscore

0.0476

1

0.8273

chcond1

0.6906

1

0.4060

chcond2

0.0615

1

0.8042

The preceding nonproportional odds model fits slope parameters, and the model seems to overfit the data. You can obtain a more parsimonious model by specifying a subset
of the parameters to have nonproportional odds. The following statements allow the parameters for the variables in the “Linear Hypotheses Testing Results” table that have p-values less than 0.1 (actdays, agesq, and income) to vary across the response functions:

The partial proportional odds model can be written in the same form as the nonproportional odds model by letting and , so the first q parameters have proportional odds and the remaining parameters do not. The last 12–q parameters can be rewritten to have a common slope: , where the new parameters contain the increments from the common slopes. The model in this form makes it obvious that the proportional odds model is
a submodel of the partial proportional odds models, and both of these are submodels of the nonproportional odds model. This
means that you can use likelihood ratio tests to compare models.

You can use the following statements to compute the likelihood ratio tests from the Likelihood Ratio row of the “Testing Global Null hypothesis: BETA=0” tables in the preceding outputs:

Therefore, you reject the proportional odds model in favor of both the nonproportional odds model and the partial proportional
odds model, and the partial proportional odds model fits as well as the nonproportional odds model. The likelihood ratio test
of the nonproportional odds model versus the proportional odds model is very similar to the score test of the proportional
odds assumption in Output 54.17.1 because of the large sample size (Stokes, Davis, and Koch, 2000, p. 249).

Note: The proportional odds model has increasing intercepts, which ensures the increasing nature of the cumulative response functions.
However, none of the parameters in the partial or nonproportional odds models are constrained. Because of this, sometimes
during the optimization process a predicted individual probability can be negative; the optimization continues because it
might recover from this situation. Sometimes your final model will predict negative individual probabilities for some of the
observations; in this case a message is displayed, and you should check your data for outliers and possibly redefine your
model. Other times the model fits your data well, but if you try to score new data you can get negative individual probabilities.
This means the model is not appropriate for the data you are trying to score, a message is displayed, and the estimates are
set to missing.